You could equally say that the atomic model of matter is relatively recent in science. Obviously it’s different because in science older models of matter are more wrong while in math you can have two different formulations that are basically the same in correctness. Still, the 19th century was a long damn time ago. I think the Riemann definition allowing you to do limits within the reals makes it superior to one that requires you to use surreal numbers - that’s just personal opinion. Making better sense of precise definitions than of analogies, though, is the different way of thinking.
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You don’t need surreal numbers to use infinitesimals precisely, almost any non-Archimedean field extension of R will do, and of course many people used infinitesimals informally (but correctly) for centuries before Riemann’s definition. The fact that Riemann’s definition lets you characterize limits just using ordered field operations is of course important and useful.
If there is anything that has been demonstrated on this thread, it is that everyone seems to have a personal story about how they relate to math that’s supposed to illuminate somehow. There is definitely a folklore of math pedagogy.
That’s how you end up with dumb books like this:
http://www.amazon.com/dp/1620970686/ref=cm_sw_r_tw_awdm_x_dE1UxbHWBPPMS
Speaking of my own experiences with being taught maths, which was pretty fucking horrible, I can squarely blame the teachers.
But the subject can be pretty unrelatable too. And if you want kids to love a subject enough to motivate them, get 'em early and teach it with passion. Seems to me that you’ve got all these cool algorithms and stuff floating around these days. So are teachers using computer games like Pokemon to teach maths now? Got to be better than “beat the crap out of them until they learn”, anyway. 
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Yeah, I brought up the surreals just because I find math naming of sets of numbers hilarious.
Using things informally but correctly usually means occasionally using them informally and incorrectly which is why math took such a hard turn to formality. I mean, what does it even mean to use something correctly? People used phlogiston theory correctly in that they managed to make accurate predictions about the world with it. By today’s standards, saying something that is true without a formal proof isn’t really being correct at all. Like all those highschool teachers who got mad at students for “cross multiplying” with equations. I can easily formally prove that the operation of “cross multiplying” is a valid one, but in the absence of that proof, getting to the right answer isn’t good enough.
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Most of them make perfect sense. For example, “rational” numbers are ratios. The terms “real” and Imaginary" were due to Descartes and had to do with roots of polynomials; a real root was one that corresponded to a length, an imaginary one is one that does not. “Surreal” on the other hand was an intentional joke by Knuth.
There’s really not much evidence of the former, excepting one noteworthy wrong result of Cauchy concerning absolute convergence. As for the latter, I agree, it is useful to have a correct foundation for the mathematics we do, however few people who use mathematics really care that much about these foundations; for example physicists are generally quite happy to use infinitesimals (the informal kind) as well as other fictions without addressing foundational existence.
I’ve had some very bad math teachers because they were in general ill equipped.
In those classes I had to do the actual teaching in class because they had no grasp of the material.
That was high school. In college I had much better math teachers and ended up taking an elective stats class just because I enjoy math.
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Mine weren’t ill-equipped, they were just human garbage.
Didn’t get that far. Wasn’t an option anyway.
Getting off-topic though. Really want to know if teachers are teaching maths via computer games (proper ones, not edutainment rubbish) because stuff like Pokemon or even MTG has got some pretty complex stuff going on at it’s heart in a way that’s “useful” to kids. Even though a maths teacher teaching Pokemon is almost guaranteed to suck all the joy from it, LOL. There’s a load of real-world stuff that can be shown to inspire passion for a subject, then jump into the dry and dusty stuff.
The learning languages thing came up earlier on. Is reciting lists of irregular verbs the best way to learn a language or is using it in conversational situations?
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(Please forgive me if I’m misinterpreting you. 
) As a programmer, I can definitely appreciate the problem of methods that get the right result (mostly) but shouldn’t be used that way because they have glaring edge cases.
My method wasn’t any different (functionally identical), it was just a way of understanding the mechanism. I regret not writing it down but I was pretty young so I can’t give myself too hard a time about it. I had no idea they were about to break my understanding of integers.
No one gave me a hard time about my method. They just tabula rasa’d my more intuitive understanding and replaced it with a confusing kludge. Yes, their way of describing the operation of basic math with positive and negative integers is (was at the time I learned anyway) a kludge.
I don’t think they’re trying to do kids bad (except the ones who do the trick question game or pretend to be Professor Snape). It’s like watching someone program file handling from scratch not because they have an actual improvement but because they don’t know about the built-in functions.
And then someone came along and wrote documentation for the above and now it’s inviolate math teaching law. Occasionally someone comes along with a simpler way of doing long division and they add it but the kids don’t understand how it works (any more than they understood the old method) so as soon as they’re out of school, they forget that too.
I don’t have a problem with math done the right way. I have a problem with math teaching done the “right” way. I have exactly as many formulas memorized today (none) as most people my age (mid-30s) and I get on great because I taught myself how to make basic algebra formulas.
Re-writing my formulas every time I need to do something with them is pretty inefficient (I’ll run out of time on any math test I’m qualified to take) but since I understand how to make them, I always have them with me. Which is something most non-mathematicians can’t say.
[spoiler]For context, I also can’t learn programming from a book. If I have a project I want to accomplish, I’ll learn the programming by doing it. And the book will help if it’s any good at all. I may not remember the exact name of whatever crucial function I learned that day was but I’ll remember it was there and be able to find it again without my old source to fall back on.
I may be mathematically inclined. I don’t know. I’m betting my learning style for programming and math is common enough.[/spoiler]
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I think it’s hard to actually offend me on this topic, and if I’m misunderstood I’m happy to take the blame for being unclear. I freely admit I’ve not been writing as well on this topic as I would like. In any case I think you’ve got me right, in so far as I’m actively defending math teachers.
But what I’ve really been trying to do, with very limited success, is shake up people’s conceptions and beliefs about math pedagogy. In particular I’m noticing how almost to a person, there is a strong tendency to generalize from particular personal experience. “This is the point where math turned on me.” Or “This was where it all clicked.” I just don’t think that with this variety of experiences and often very old (i.e. unreliable) memories, we can move the discussion forward. I have my own “mathematical moments,” but I realize now that they’re increasingly meaningless. They happened decades ago and as time wears on, as diaries and writings will attest, my memory of myself from that period grows increasingly unreliable.
In fairness to you, I think that applies to everyone to some extent. It’s a big reason why I have a hard time knuckling under to learn C/C++. I do want to learn it, but I can’t think of anything I want to use it for that is level-appropriate and that tends to sap my motivation a little.
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I fully admit that I suck at maths. I’ve been very candid about this here. That said, it’s not something I’m proud of. It’s like saying “lol I can’t read. I’m so cool”.
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Exactly. The social stigma for being illiterate is high, whereas many will throw out “Oh, I’m bad at math” as if it’s no big deal.
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Welcome to my world. Any time someone asks me what I do, and I answer, they tell me how bad they were at math. Then they apologize, as if it was somehow offensive to me. (Which it is).
Except once. A dentist, who told me (while drilling) that he used to teach statistics in the Navy. Another time he told me about his father’s collection of human body parts, including brains, in formaldehyde. Which he’d inherited and still had. He was interesting. And a great dentist.
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I assume they are using MTG as a teaching tool in PhD programs. Possibly also in law school.
I can isolate that point, it was when I started to see all mathematics as propositions P1 … Pn and NAND gates. My experience may have been different than most.
The next time someone says, “I always hated math” I’m going to be very tempted to say, “I’ve always hated people.” And the only reason I haven’t said it before is because I just thought of it this week.
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My stock answer for people who say they hated chemistry is, “Oh good. Job security.”
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That’s basically what I said to people who apologized for not being good with computers, when I had a helpdesk job.
“Don’t worry about it. If everyone was as good with computers as I am, they wouldn’t need a helpdesk, and I wouldn’t have a job.”
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I say that too. But I know it’s a lie. Most of the stuff I fix for people is solved with a reboot, or even better Logging out then logging back in.
My standard line when people say stupid shit like “you’re amazing” or “you’re a wizard” or whatever is “Naw, I’m just a dude who went to college and got a degree in this. You’re someone who designs clothes/sells clothes/manages people/invents and cooks delicious food/tabulates accounts/sources materials/etc. I can’t do that, and I’d hate it if I had to.”
What I really want to say to most of these people is “Everything you do today will have a computer involved in one way or another. It’s irresponsible of you to not know at least the basics.” But I’d be fired so fast if I expressed that on any of my calls.
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I’ll just leave this here, because if I ever get a tattoo, it’ll be this:
Euler’s Identity is the most beautiful and elegant piece of math I’ve ever come across, and I still can’t quite understand it.
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I said somewhere that Euler’s Identity is the most fun argument that this universe has a creator. Because that’s been put there solely to mess with our heads.
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Seriously, I had a difficult time getting ahold of Pi until I was given a bunch of practical examples and learned what transcendental numbers are. Then they threw e at me, and I can’t say I’ve ever understood how we got to e, but it works perfectly. i was easy enough, since imaginary numbers aren’t really any different from vector math. It’s just adding an extra dimension to the graph, and I can do dimensions till the cows come home. So, really I think the mysterious part for me comes from the fact that I don’t know how you multiply i with Pi like that and use it to square e which I don’t really understand at all.
Coincidentally my favorite movie is about math.
π: Faith in Chaos.
Pi is also the first film Darren Aronofsky directed. He went downhill fast with Inception in my opinion.
I don’t hate Inception, but I find the material it covered was philosophically juvenile. Anyone who’s thought seriously has come up against solipsism, and we’ve found our ways out of it, by necessity. Those who haven’t are plagued with mental illness, so to try and introduce it as a new concept to a wide audience seems rather passé and kind of intellectually insulting to me.
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